Damped and forced oscillations

A block of mass 0.1 kg is connected to an elastic spring of spring constant 640 Nm$^{-1}$ and oscillates in a damping medium of damping constant $10^{-2}$ kg s$^{-1}$. The system dissipates its energy gradually. The time taken for its mechanical energy of vibration to drop to half of its initial value, is closest to :

An LCR circuit is equivalent to a damped pendulum. In an LCR circuit the capacitor is charged to $Q_0$ and then connected to the L and R as shown below :

If a student plots graphs of the square of maximum charge ($Q^2_{\text{Max}}$) on the capacitor with time(t) for two different values $L_1$ and $L_2$ ($L_1>L_2$) of L then which of the following represents this graph correctly ? (plots are schematic and not drawn to scale)

A simple harmonic oscillator of angular frequency 2 rad s$^{-1}$ is acted upon by an external force $F=\sin t$ N. If the oscillator is at rest in its equilibrium position at $t=0$, its position at later times is proportional to :

A pendulum with time period of 1s is losing energy due to damping. At certain time its energy is 45 J. If after completing 15 oscillations, its energy has become 15 J, its damping constant (in s$^{-1}$) is :

The amplitude of a simple pendulum, oscillating in air with a small spherical bob, decreases from 10 cm to 8 cm in 40 seconds. Assuming that Stokes law is valid, and ratio of the coefficient of viscosity of air to that of carbon dioxide is 1.3, the time in which amplitude of this pendulum will reduce from 10 cm to 5 cm in carbondioxide will be close to ($\ln5=1.601$, $\ln2=0.693$).

The angular frequency of the damped oscillator is given by,
$$\omega=\sqrt{\left(\frac km-\frac{r^2}{4m^2}\right)}$$
where k is the spring constant, m is the mass of the oscillator and r is the damping constant. If the ratio $\dfrac{r^2}{mk}$ is 8%, the change in time period compared to the undamped oscillator is approximately as follows :

Consider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is